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Differentiation — chain, product and quotient rules, related rates (IB Math AI HL)
Updated October 2026
Quick answer: HL AI uses the chain, product and quotient rules with the derivatives of xⁿ, sin x, cos x, tan x, eˣ and ln x. In a related rates question, write an equation linking the quantities, differentiate it with respect to time, and substitute the values at the moment asked about.
HL AI extends SL AI with the chain, product and quotient rules and related rates. The 'water tank filling' family of questions turns on d/dt applied to a relationship between variables.
Practise Differentiation — chain, product and quotient rules, related rates with AI marking
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Chain rule: dy/dx = (dy/du) × (du/dx). For example, d/dx(e3x²) = 6x e3x².
Product rule: d/dx(uv) = u dv/dx + v du/dx. Quotient rule: d/dx(u/v) = (v du/dx − u dv/dx) ÷ v². Both are in the formula booklet.
Related rates: dA/dt = (dA/dr) × (dr/dt). Find the rate you know, the rate you want, and the formula that links the two variables.
Substitute the particular value (such as r = 10) only after differentiating; substituting first turns a variable into a constant.
The second derivative tells you the concavity, and a sign change of f″(x) gives a point of inflection.
Units of a rate are units of the quantity per unit of time: m² per minute, cm per second.
Worked example
Oil spreads on water in a circle. The area increases at a constant 30 m² per minute. Find the rate at which the radius is increasing when the radius is 10 m, and the rate at which the circumference is increasing at that moment.
Link the variables: A = πr², so dA/dr = 2πr.
Chain rule: dA/dt = 2πr × dr/dt.
At r = 10: 30 = 2π × 10 × dr/dt, so dr/dt = 30 ÷ (20π) = 3/(2π) ≈ 0.477 m per minute.
Circumference: C = 2πr, so dC/dt = 2π × dr/dt = 2π × 3/(2π) = 3 m per minute.
Answer: dr/dt = 3/(2π) ≈ 0.477 m per minute; dC/dt = 3 m per minute.
Write down the rate you are given and the rate you want, both as derivatives with respect to time. Find a formula linking the two quantities, differentiate it, use the chain rule to connect the rates, and only then substitute the values at the moment in the question.
Where can I practise differentiation questions?
On the Advanced Differentiation and Related Rates questions page: 40 exam-style HL AI questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for differentiation?
Yes: a GDC is allowed on every HL AI paper, so practise the calculator method as well as the working.